2021 VJC P2 Q10

In this question you should state clearly all the distributions that you use, together with the values of the appropriate parameters.

A certain bakery bakes two types of cookies: butter cookies and chocolate cookies. The masses of butter cookies have the distribution \(N\left( 15,\ {{0.4}^{2}} \right)\) and the masses of chocolate cookies have the distribution \(N\left( 20,\ {{1.2}^{2}} \right)\). The units for mass are grams.

1. Find the probability that the mass of a randomly chosen butter cookie is more than \(15.5\) grams.[1]
2. \(10\) butter cookies are randomly chosen. Find the probability that at least \(4\) of them each has mass more than \(15.5\) grams.[3]

The cookies are sold by weight. Butter cookies cost \(\$6\) per \(100\) grams and chocolate cookies cost \(\$7.50\) per \(100\) grams.

3. Miss Lee bought \(12\) butter cookies and \(12\) chocolate cookies for her family. Find the probability that she paid less than \(\$29\).[4]
4. State an assumption needed for your calculations in part (iii).[1]

The waiting time, \(T\) minutes, before a customer is served at the bakery has a mean of \(16\) minutes and a standard deviation of \(9\) minutes.

5. Give a reason why a normal distribution, with this mean and standard deviation, would not give a good approximation to the distribution of \(T\).[1]
6. The waiting times of \(n\) randomly chosen customers at the bakery are taken, where \(n>30\). Given that the probability that the average waiting time of these \(n\) customers is between \(16\) minutes and \(18\) minutes is more than \(0.48\), estimate the least value of \(n\).[5]